Average Error: 3.5 → 2.1
Time: 4.9m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.794146793469779 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2} \cdot \frac{1}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)} \cdot \left(\left(\alpha + \beta\right) \cdot 0.25 + 0.5\right)\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.794146793469779e+183

    1. Initial program 1.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified1.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
    3. Using strategy rm

    4. Applied div-inv1.4

      \[\leadsto \frac{\color{blue}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    if 2.794146793469779e+183 < alpha

    1. Initial program 16.5

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified16.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
    3. Using strategy rm

    4. Applied div-inv16.5

      \[\leadsto \frac{\color{blue}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity16.5

      \[\leadsto \frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)} \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{\color{blue}{1 \cdot \left(\left(1.0 + \left(\beta + \alpha\right)\right) + 2\right)}}\]

    7. Applied times-frac16.8

      \[\leadsto \color{blue}{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{1} \cdot \frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]

    8. Taylor expanded around 0 6.4

      \[\leadsto \frac{\color{blue}{0.25 \cdot \alpha + \left(0.25 \cdot \beta + 0.5\right)}}{1} \cdot \frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    9. Simplified6.4

      \[\leadsto \frac{\color{blue}{0.5 + 0.25 \cdot \left(\alpha + \beta\right)}}{1} \cdot \frac{\frac{1}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.794146793469779 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2} \cdot \frac{1}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)} \cdot \left(\left(\alpha + \beta\right) \cdot 0.25 + 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))